332
IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998
Time-Domain Image Processing
Using Dynamic Holography
Y. Ding, David D. Nolte, Michael R. Melloch, Senior Member, IEEE, and Andrew M. Weiner, Fellow, IEEE
Abstract—Coded ultrafast optical pulses can be treated as onedimensional (1-D) images in the time domain. We have converted
space-domain images into time-domain images using diffraction
from dynamic holograms inside a Fourier pulse shaper, with
photorefractive quantum wells (QW’s) used as the dynamic
holographic medium. We present several examples, in which
amplitude or phase modulation of the hologram writing beams
modifies the complex spectrum of the femtosecond output, resulting in a time-domain image. Both storage and processing of
time-domain images can be achieved, depending on the hologram
writing geometry and power densities. Time-domain processing
operations such as edge enhancement, Fourier transform, and
correlation are demonstrated.
Index Terms—Holography, image processing, nonlinear optics,
optical pulse shaping, optical signal processing, quantum-well
devices, ultrafast optics.
I. INTRODUCTION
B
Y ANALOGY with function generators and signal processors in electronics, the ability to arbitrarily manipulate
ultrafast pulses is of great interest and practical importance
in optical communication and information processing. Femtosecond (fs) lasers open possibilities to exploit the large
bandwidth of optics (several terahertz (THz) for commercially
available sub-100-fs lasers). As one method to utilize this
bandwidth, ultrafast pulses can be shaped and coded using
Fourier synthesis techniques. Shaping of laser pulses on the
femtosecond time scale was demonstrated in 1988 [1], in
which the optical frequency components of a femtosecond
pulse were manipulated by an amplitude or phase mask in the
Fourier domain. By changing the mask, different shaped pulses
were obtained. Extensive work has been done in this area
[2] using different kind of masks such as photolithographic
amplitude and phase masks [1], [3], dynamic liquid crystal
spatial light modulators [4]–[7], accustooptic light modulators
[8], computer-generated holographic plates [9], and dynamic
holograms [10], [11]. Femtosecond pulse-shaping experiments
using photorefractive quantum wells (PRQW’s) [11] have
demonstrated real-time control over the hologram with the
material response time in the microsecond range.
Manuscript received October 1, 1997; revised February 24, 1998. This work
was supported by the U.S. Department of Defense Focused Research Initiative
through U.S. Air Force Office of Scientific Research under Grant F49620-951-0533, by the National Science Foundation under Grant 9 708 230-ECS and
by the Rome Laboratory under Grant F30602-96-2-0114.
Y. Ding and D. D. Nolte are with the Department of Physics, Purdue
University, West Lafayette, IN 47907-1396 USA.
M. R. Melloch and A. M. Weiner are with the School of Electrical and
Computer Engineering, Purdue University, West Lafayette, IN 47907 USA.
Publisher Item Identifier S 1077-260X(98)03771-X.
Shaped pulses carrying information may have complicated
temporal patterns which can be treated as one-dimensional
(1-D) images in the time domain. The term “time-domain
image” is most appropriate for cases when the shaped forms
are obtained by converting space-domain images into the time
domain. Time–space conversions are useful in femtosecond
optical signal processing, optical transmultiplexing, image
processing, and temporal pattern generation and recognition.
Both time-to-space and space-to-time conversions have been
demonstrated [12]–[16], and time-domain images have been
obtained using both fixed and dynamic Fourier holograms
[9], [17]. Processing time-domain images [17] (by analogy
with processing conventional space-domain images) may open
new possibilities for applications in optical communication,
signal processing and data storage. Possible operations for
time-domain image processing include: 1) single-image processing, such as noise suppression, defect correction, edge
enhancement, time reversal, etc., where there is only one image
involved, and 2) multi-image processing, such as convolution
or correlation, where two or more images are involved. By using spectral holography [18], operations such as time reversal,
correlation and matched filtering of time-domain femtosecond
input optical pulses have been realized [19], [20]. Nonlinear
spectral holography permits more elaborate signal-processing
operations [21]. In this paper, we discuss in detail spaceto-time conversion using PRQW’s for dynamic holography
within a femtosecond pulse shaper and demonstrate the possibility of performing various processing operations, such as
Fourier transform, edge enhancement, and correlation, as part
of space-to-time conversion.
The second section of this paper explains the basic principle
we used for time-domain image generation and processing, i.e.,
space-to-time conversion and processing during conversion,
and describes the experimental setup, followed by a section
demonstrating the suitability of PRQW for femtosecond timedomain image processing applications. The fourth and the
fifth sections present results in the generation and processing
of time-domain images, where we will give examples of
single- and multi-image processing operations such as edge
enhancement and correlation.
II. PRINCIPLES
AND
EXPERIMENTAL SETUP
Ultrafast optical pulses contain a wide bandwidth of, for
example, 4 THz for a typical 100-fs pulse duration with a
Gaussian shape, corresponding to 10 nm at a center wavelength
of 850 nm from a Ti:sapphire laser. Time-domain images can
be generated and processed in the frequency domain using
1077–260X/98$10.00 © 1998 IEEE
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DING et al.: TIME-DOMAIN IMAGE PROCESSING USING DYNAMIC HOLOGRAPHY
333
Fig. 1. Experimental setup for femtosecond time-domain image generation and processing using dynamic holograms written in photorefractive QW’s,
consisting two planes perpendicular to each other: pulse-shaper plane and hologram writing plane. The focusing length of the cylindrical lenses in the pulse
shaper is 10 cm and the two gratings have a groove density of 1800 lines/mm, leading to a spatial spectrum separation of 0.35 mm/nm.
Fourier synthesis techniques. As an ultrafast optical pulse with
a wide spectrum is dispersed by a dispersive element like a
grating, its spectral components are spatially spread. If these
components are filtered and modified by an amplitude or phase
mask and then recombined, the resultant temporal output,
which is the Fourier transform of the modified spectrum,
contains the information of the mask. If the amplitude/phase
mask is itself the Fourier transform of a certain space-domain
image, for example a square aperture, the resultant temporal
output envelope will take the same form as the space image,
resulting in a time-domain image (a square wave). Any deviation of the mask from the Fourier transform of the space
image will “distort” the converted time-domain image. Such
deviation can be intentional and can be used to manipulate the
time-domain images. The mask in the Fourier plane can be
either a physical mask or a hologram. In our experiments, the
mask is a dynamic hologram written in a PRQW by CW laser
beams, which allows us to modify the output spectrum in situ.
The temporal output is the Fourier transform of the frequency
components diffracted from the hologram. By modifying the
hologram writing beams, for example by projecting the Fourier
transform of a space image onto the hologram medium by
using a simple lens, time-domain images are generated by
space-to-time conversion. The time-domain images can take
different forms, if we control the parameters such as writing
beam power densities, lens positions, etc.
The experimental setup is shown in Fig. 1 with main
parameters in the figure caption. It is a dispersion-free 4fsystem allowing the synthesis of femtosecond pulses. The
setup is similar to the conventional pulse-shaping apparatus [1]
(pulse shaper) with two differences fitting the use of dynamic
holography: first, a PRQW is placed in the Fourier plane in
the pulse shaper instead of a mask, and second, the lenses
used in the pulse shaper are cylindrical lenses instead of the
conventional spherical lenses. The hologram is written in the
PRQW by two cw beams from either a diode laser operating at
685 nm or a He–Ne laser operating at 633 nm. The power of
each writing beam is 2–7 mW. By using cylindrical lenses
in the pulse shaper, the incident femtosecond beam at the
Fourier plane remains unfocused in the vertical direction, so
that its power density will be not so high as to erase the
hologram, leading to optimal photorefractive diffraction. The
Fourier transforms of the femtosecond probe beam and the
diffracted beam by the cylindrical lenses are, therefore, 1-D
transforms in the horizontal direction.
The femtosecond source is a mode-locked Ti-sapphire laser
with a nominal pulsewidth of 100 fs and a central wavelength
adjusted to the excitonic absorption line of the QW’s. The
laser pulses are characterized to be transform-limited and
have a Gaussian shape by interferometric second-harmonic
generation (SHG) autocorrelation, power spectroscopy, and
spectral interferometry. The femtosecond probe beam with
a power of 2.6 mW and a diameter of 1.8 mm is incident
on the first grating of the pulse shaper, where it is spectrally
dispersed. The frequency components are diffracted from the
hologram in the PRQW at the Fourier plane. The diffracted
beam travels in the vertical plane determined by the two writing beams. The diffracted optical components are recombined
by the second cylindrical lens and by the grating, forming the
femtosecond output carrying the information of the hologram.
The spectrum of the femtosecond output is captured through
a spectrometer with a CCD array, and the temporal shape of
the pulse is measured by electric-field cross correlation with
an unshaped transform-limited reference pulse. Electric-field
cross correlation is a technique suitable for weak ultrashort
pulse characterization [22], [23], where the field-interferogram
of the signal and reference pulses is recorded by scanning
the time delay between the two pulses. For space-to-time
conversion or single-image processing, a lens is put into one of
the writing beams, called the image-carrying beam, to project
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998
a mask or its Fourier transform onto the PRQW, and the
reference beam is unaltered. In the experiments of multi-image
processing, both writing beams contain a lens. The image
Fourier plane is 22 off the Fourier plane of the pulse shaper,
for technical simplicity. This has no detrimental effects on our
results because of the long focal length of the lenses in the
writing beams (100–200 mm). Neutral density filters are used
to adjust the power densities of the two writing beams.
The pulse shaper is carefully aligned to be dispersion
free with no PRQW in it, giving identical output pulses
As PRQW is
to the incident ones. Then a GaAs–Al Ga
placed in the Fourier plane. Two samples were used in our
0.1 and the other with 0.3. The
experiments, one with
heavy-hole excitonic absorptions are centered at 847 and 836
nm, respectively. Titanium-gold contacts were evaporated on
10 mm aperture. Across the 1-mm
the samples with a
gap, a field of 3–7 kV/cm is applied in the plane of the
QW’s, perpendicular to the growth direction (transverse field
geometry [24], [25]). The angle between the two writing beams
is 3.3 degrees and the output diffraction efficiency is on the
order of 0.1%.
III. DIFFRACTION OF FEMTOSECOND PULSES FROM
PRQW’s IN THE TRANSVERSE-FIELD GEOMETRY
As a dynamic holographic medium, PRQW’s have high
)
nonlinear sensitivities (refractive index change
with small driving power densities ( 10 W/cm ) and fast response times ( 10 s) [24], [25], [26]). The holograms in such
materials are thin and the diffraction is in the Raman–Nath
regime with no Bragg-matching necessary, providing much
larger space-bandwidth products than bulk materials [27],
which is important for Fourier filtering in image processing applications. PRQW’s can also be designed using semiconductor
bandgap engineering to match different application wavelengths, such as 1.55 m for optical fiber communications [28].
For dynamic Fourier manipulation and processing of timedomain images, an ideal holographic material should have
a flat amplitude and phase response over the bandwidth
of the pulses. The diffraction from the hologram in such
a material would not distort the probe pulse shape if the
hologram itself is not modified by any masks. Photorefractive holograms in PRQW are based on the electroabsorption (electric-field-induced absorption) and electrorefraction
(electric-field-induced refractive index change) of quantumconfined excitons. This effect has a typical bandwidth of 5 nm
[29], which limits the 10-nm bandwidth for a typical 100-fs
Gaussian pulse at 850 nm and broadens the diffracted pulses
by a factor of two. Most importantly, the effect by itself
is strongly wavelength dependent. The oscillatory behavior
as a function of wavelength might be expected to lead to
severe distortion of the diffracted pulse shape. However, it
was experimentally observed that both spectrum and temporal
shape of the diffracted pulse appear to be nearly Gaussian with
a time-bandwidth product of 0.43, which is very close the that
for Gaussian pulses (0.44), although the limited bandwidth
of the quantum-confined excitonic transitions broadens the
diffracted pulses [29]. This implies that the diffracted pulse is
Fig. 2. Phase of the diffracted pulse from a 10% Al-barrier PRQW in the
pulse shaper. The dashed line represents the phase extracted from the spectral
interferometry pattern (solid line) between the reference pulse and diffracted
pulse and the dot line is the phase calculated from the electroabsorption and
electrorefraction data.
nearly transform-limited and the phase of the diffracted pulse
in the frequency domain is linear.
To understand the linear phase, the field amplitude of the
first-order diffraction of a femtosecond incident pulse
from a photorefractive grating in PRQW is given by [29]
(1)
with the transmission function
(2)
and the diffraction function
(3)
where is the complex refractive index, is the interaction
length and is the wavevector in vacuum. As long as the
frequency dependence of the background absorption is not
strong, which is the case when the excitonic absorption line is
broadened by the applied electric field in the transverse-field
geometry, where the electric field is parallel to the QW layers,
is mainly determined
the phase of the diffracted field
in (3)
by the phase of the diffraction function
(4)
indicating that the phase is given by the ratio of the electroreand the electroabsorption
, which are
fraction
related by the Kramers–Kronig relation [30]
(5)
where stands for the Cauchy principal value and is the light
velocity. Using the electroabsorption measurement
and the calculated electrorefraction data
and (4), we
calculate the phase of the diffracted beam in the frequency
domain as shown in Fig. 2. The phase of the diffracted pulse
can be measured experimentally using spectral interferometry
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DING et al.: TIME-DOMAIN IMAGE PROCESSING USING DYNAMIC HOLOGRAPHY
[31], which is a versatile linear technique for determining
both phase and shape of weak femtosecond pulses, where the
spectrum of a reference pulse interferes with the spectrum of
the diffracted pulse with a chosen time delay. Fig. 2 shows
the spectral interferometry pattern with a time delay of 1500 fs
and the phase of the diffracted pulse extracted from it, together
with the phase calculated from (4) using electroabsorption and
electrorefraction data. After subtracting a linear part of the
curves, the phase is quite flat around the center wavelength of
847 nm for the PRQW sample used.
The linear phase in the diffracted pulse is a result of
the character of the transverse-field PRQW devices, where
electroabsorption is caused by the broadening of the transition lineshape. The characteristic electroabsorption spectrum
is approximately the second derivative of a Gaussian transition lineshape [24], which is an even function.
is related to
by the
The electrorefraction
Kramers–Kronig relation, which transforms an even function
is
to an odd one. Therefore, to first order the phase
linear in frequency.
We assume an excitonic absorption line that is Gaussian at
room temperature [32]
(6)
is the FWHM of the lineshape,
is the
where
is the ionic plasma frequency, and is
Planck constant,
the oscillator strength.
For the transverse-field geometry at relatively small fields
(typically 15 kV/cm), the linewidth becomes
with the linewidth without field . The electroabsorption is
then
(7)
The maximum of the absolute electroabsorption
with the value of
the center frequency, i.e.,
is at
(8)
Near the central frequency of the excitonic absorption,
is approximately a constant because the linewidth is much
smaller than the central frequency, and (7) can be written as
(9)
, it can be further rewritten as
Using
335
The Kramers–Kronig relation transforms the cosine functions to the sine functions. Therefore, the electrorefraction
from (5) and (10) is
(11)
After some algebra, it can be written as
(12)
From (9) and (12), the phase of the diffraction function
in (3) is then
(13)
which is a linear function with . A linear phase in the
frequency domain introduces only a delay of the output pulse
in the time domain but not distortion. The slope in (13) is
inversely proportional to the linewidth and is independent of
the applied field.
The amplitude of the diffraction function can also be deduced from (9) and (12), which is proportional to
. It is easily seen that
takes the same
Gaussian shape as the absorption, which determines the bandwidth limit of the PRQW. Therefore, the output diffracted
pulse is bandwidth-limited with a Gaussian shape, as experimentally observed. This distortionless quality makes the
PRQW’s suitable for use in pulse manipulation.
IV. GENERATION OF TIME-DOMAIN
IMAGES BY SPACE-TO-TIME CONVERSION
In this section, we discuss the generation of time-domain images by space-to-time conversion using holographic diffraction
from a PRQW, where a conventional image in the spacedomain is converted dynamically into the time domain. Using
the experimental setup in Fig. 1, we realize the space-totime conversion by projecting the Fourier transform of a
conventional image onto the PRQW in the pulse shaper. If
the hologram amplitude in the PRQW is proportional to the
Fourier-transformed image, the shaped temporal pattern, which
is the Fourier transform of the diffracted spectrum from the
hologram, should take on the same shape as the input spacedomain image.
The interference pattern of the two coherent writing beams
has a modulation depth of
(10)
where we neglected the
- and higher-order terms.
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(14)
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998
with field amplitude
(power density ) of the reference
writing beam (spatially homogenous), those of the image,
, and of the femtosecond probe
carrying beam
,
, respectively. is a factor describing the
beam
ratio of the photocarriers induced at the probe and writing
wavelengths. Because the femtosecond probe beam has a
much longer wavelength (850 nm) than that of the writing
beams (633 or 685 nm), it was shown that the erasure of the
grating by the probe beam is much weaker than the effect of
) due to the difference
the writing beams [33] (i.e.,
in the photoconductivity and interaction length at the two
wavelengths. From (14), depends only on the power density
and has the maximum value of
ratio of the two beams
(neglecting erasure). The hologram written
unity when
as long as
is small.
in the PRQW is proportional to
approaches unity, the amplitude of the photorefractive
As
grating, and therefore the diffracted electric field amplitude
[34], [35]. In holographic
as well, become sublinear with
recording, the hologram pattern is a faithful replica of the
original electric field distribution only if the power density of
the image-carrying beam is much less than the power density
of the reference beam and the material response is linear in
and
does not
power density. In this case (
, and the hologram has the same
depend on ),
.
distribution as the image-carrying beam
As an example, a single slit with a width of 380 m is used
as a space-domain image and is put into the image-carrying
beam. This image is Fourier transformed onto the PRQW by a
150 mm, performing
cylindrical lens with a focal length of
the Fourier transform one-dimensionally. The power density
distribution at the Fourier plane takes the form of a squared
function when
(15)
, writing wavelength , and the coordinate
for a slit width
in the Fourier plane of the Fourier lens. Using
380
m,
150 mm, and
685 nm, the FWHM of the
main maximum of the distribution in (15) is approximately
250 m. The power density of this maximum is, therefore,
not more than twice as large as the power density before the
Fourier imaging, which is approximately equal to the reference
power density. Neutral density filters are therefore used to
reduce the image-carrying beam power density by an order
of magnitude to ensure that the reference beam has higher
power density than the image-carrying beam everywhere. The
measured diffracted power spectrum from the hologram also
takes the sinc squared distribution as shown in Fig. 3(a). For
comparison, the theoretical curve using (14) and (15) is also
given, where the coordinate is translated into wavelength
using the spectral dispersion of the pulse shaper of 0.35
mm/nm. The PRQW spectral response with a power density
full-width at half-maximum (FWHM) of 4.2 nm for the 30%
Al-barrier sample is also included. The spectrum agrees well
with the simple theory. The finite spectral resolution of both
the pulse shaper and the measurement apparatus, and the angle
(a)
(b)
Fig. 3. (a) Spectrum and (b) the electric-field cross-correlation data and
the extracted envelope of the diffracted temporal image when the Fourier
transform of a single slit is projected onto the PRQW. The reference beam
power density is higher than the image-carrying beam power density. This
shows the space-to-time conversion. The corresponding theoretical calculation
10 is also given in (a).
of the spectrum at
between the image Fourier plane and the Fourier plane in the
pulse shaper cause the minima of the power density distribution in the experimental data to remain finite. The temporal
shape of the diffracted (or reconstructed) femtosecond outputs
is measured by electric-field cross correlation as shown in
Fig. 3(b), which corresponds to the input space-domain image;
a square-wave packet. It is important to note that Fig. 3(b)
contains the electric-field cross-correlation data. The side lobes
would appear much smaller if the power density rather than
the field amplitude were plotted.
V. PROCESSING OF FEMTOSECOND TIME-DOMAIN IMAGES
The images can be manipulated as they are converted
from the space-domain to the time domain by changing the
hologram writing power density, writing geometries and the
relation between the input images, etc. Single image processes
as well as multiple image processes can be realized.
A. Single-Image Processing: Edge Enhancement, Fourier
Transform, Noise Suppression, and Time Reversal
Many applications require images with enhanced features,
such as edges and noise, or certain transformations of a image,
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DING et al.: TIME-DOMAIN IMAGE PROCESSING USING DYNAMIC HOLOGRAPHY
(a)
337
(b)
(c)
Fig. 4. (a) Spectrum and (b) the electric-field cross-correlation data and the extracted envelope of the diffracted temporal image when the Fourier transform
of a single slit is projected onto the PRQW. The reference beam power density is lower than the maximum image-carrying beam power density. This
shows edge enhancement of the converted square wave. The corresponding theoretical calculations of the spectrum and the temporal field envelope at
0.01 are also given in (a) and (c).
such as Fourier transform and time reversal. These are singleimage processing operations.
Edge Enhancement of a Time-Domain Image: Using the
experimental setup in Fig. 1, we can perform edge enhancement on the converted time-domain image by choosing an
appropriate hologram-writing power density ratio. We know
from (14) that the hologram in the PRQW depends only on the
. For the hologram
power density ratio of the two beams
pattern to be a faithful replica of the original electric field
distribution, the reference beam power density must be higher
than the most intense spot of the Fourier transform of the
image. When this condition is not satisfied, the hologram
amplitude saturates and even decreases at positions where
the reference beam power density is weaker than the imagecarrying beam. Such saturation and decrease can serve to
enhance the high-frequency optical frequency components at
the expense of the strong peak at the fundamental frequency.
This produces a high-pass filter which can lead to edge
enhancement in the reconstructed images. Such effects were
demonstrated in the space domain [36], [37] and used to detect
dust and other small defects in periodic photolithographic
masks [38].
Experimentally, the power density of the reference beam
is reduced by two orders of magnitude relative to the image-
carrying beam using neutral-density filters. As a result, the interference of the two writing beams (the hologram) washes out
the primary maximum in the image-carrying beam and extracts
the higher spatial frequency components of the slit-diffraction
[sidelobes with lower power in Fig. 3(a)], which describe
edges or other suddenly changing features. The diffracted
spectrum from such a hologram is shown in Fig. 4(a), together
with the theoretical calculation. The temporal shapes of the
diffracted (or reconstructed) femtosecond outputs, which are
the Fourier transform of the electric field spectrum, are given
in Fig. 4(b), showing the enhanced edges when compared
with the time-domain image in Fig. 3(b). The side wings
in Fig. 4(b) are also expected from the theoretical Fourier
transform of (14), which are shown in Fig. 4(c). Physically, the
high-pass filtering operation in this experiment transforms the
rising and falling edges of the input image into the observed
derivative-like features.
Fourier Transform of a Time-Domain Image: The output
time-domain image is the Fourier transform of the diffracted
spectrum from the hologram. If the hologram corresponds to
the space-domain image itself, then the resulting time-domain
image is related to the Fourier transform of the space image.
Experimentally, this is realized by directly imaging a mask
(amplitude or phase) onto the PRQW in the pulse shaper.
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998
(a)
(a)
(b)
Fig. 5. (a) Spectrum and (b) electric-field cross-correlation data and the
extracted envelope of the shaped femtosecond pulse when a phase step mask
is imaged onto the PRQW, showing the Fourier transform of a phase-step
mask (double pulse).
As the first example of this processing operation, a fixed
phase mask is placed into the image-carrying beam and imaged
onto the PRQW using a spherical lens with a focal length of
200 mm. The photolithographically etched mask contains a
phase step of for a wavelength of about 620 nm. For our
writing beams from the diode laser at around 685–690 nm, the
or smaller, depending on the
phase step has a value of
incident angle of the writing beam. This phase step changes
the phase of one half of the writing beam by the amount of
with respect to the other half, giving rise to a phase step
in the hologram. The diffracted beam will experience a phase
function
(16)
with the center frequency
becomes
. The output electric field of (1)
(17)
The experimental spectrum of the diffracted pulse is shown
in Fig. 5(a). By interfering this spectrum with the spectrum
of the reference pulse having a chosen time delay (spectral
interferometry), the phase step in the femtosecond output can
be extracted, which is also shown in the figure. The measured
(b)
Fig. 6. (a) Spectrum and (b) electric-field cross-correlation data and the
extracted envelope of the shaped femtosecond pulse when a double-slit
amplitude mask is imaged onto the PRQW, showing the Fourier transform
of the double-slit.
phase step is
, which is close to the estimated value. The
notch in the spectrum is due to the destructive interference of
the frequency components focused near the mask step [39].
The output pulse in the time domain is the Fourier transform
in (17), which results in a double pulse. The
of
experimental data of the electric-field cross correlation and
the envelope extracted from the data are shown in Fig. 5(b).
In Fig. 6, we show a second example, in which a double-slit
amplitude mask is imaged onto the PRQW with an magnification of 2.2. The magnification was chosen for technical
200 mm) is introduced in
simplicity. An identical lens (
the reference beam to compensate the wavefront difference in
the two writing beams due to the magnification. The lens in
the reference beam also compensates the quadratic phase shift
caused by the single imaging lens (although it is not severe in
our experiments). An alternative would be using the canonical
– –
imaging system instead of the single lens in the
imaging beam, which in fact images the wavefronts with a
magnification restricted to 1. The slit width is approximately
250 m with a separation of 300 m. The double slit pattern at
the Fourier plane allows only two discrete frequency groups to
be diffracted [Fig. 6(a)], resulting in a double-slit interference
pattern in the time domain. In this case, the function
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DING et al.: TIME-DOMAIN IMAGE PROCESSING USING DYNAMIC HOLOGRAPHY
339
in (17) is
(18)
with the frequency separation
:
and the box function of width
(19)
in (18) gives
The Fourier transform of (17) using
a typical double-slit interference pattern, as we observed
experimentally [Fig. 6(b)], which is the Fourier transform of
the double-slit.
Using Fourier filtering, other processing operations for
single time-domain images such as noise suppression and time
reversal can also be realized. For example, time reversal can be
obtained by using an inverting telescope in the image-carrying
beam in the hologram writing geometry.
(a)
B. Processing of Multiple Images: Correlation
of Time-Domain Images
In optical processing, one of the most ubiquitous operations
is the correlation, which is essentially a method for assessing
the degree of similarity between two images. For two 1-D
and
, the correlation is defined as
images
(20)
The relationship between (20) and the Fourier transform of
the input images, known as the correlation theorem, is useful
in image processing and allows one to correlate in the time
domain by simple multiplication in the frequency domain. That
has the Fourier transform
and
has the
is, if
, then the Fourier transform of the
Fourier transform
is the correlation
:
product
(21)
represents the Fourier transform. Experimentally,
where
the product of the Fourier transform of the first image and
the phase conjugate of the Fourier transforms of the second
image are obtained by projecting the Fourier transform of
a space-domain image in each of the two writing beams
onto the PRQW using Fourier lenses. The interference term
of the power density distribution on the PRQW
is proportional to the product of the two
, which gives the distribution or envelope of
fields
the hologram modulation as long as the power density response
of the PRQW remains linear, which was proven to be true in
our experiments. The field diffraction from the hologram then
is related to the product of the Fourier transform of the first
image and the phase conjugate of the Fourier transform of
the second image. The output time-domain image (the Fourier
transform of this product) is then the correlation of the two
input images.
We performed the correlation experiments using two identical single-slits as input images, whose Fourier transform is a
(b)
Fig. 7. (a) Spectrum and (b) electric-field cross-correlation data and the
extracted envelopes of the correlation of two images (square-wave packets).
The Fourier transform of the two input images are projected onto the PRQW.
sinc function as given in (15). The diffracted power spectrum
is then proportional to the fourth power of the sinc function
in (15), as shown in Fig. 7(a). The temporal output is the
Fourier transform of this spectrum and is a triangle function,
corresponding to the correlation of two square-wave packets.
Fig. 7(b) shows the electric-field cross correlation of such a
processed time-domain image.
Another important operation is the convolution, which is
also called folding product, composition product or superpois
sition integral. The convolution of two images
given by
(22)
Similar to the correlation, the convolution theorem states
(23)
instead of
in
This is similar to (21), but with
can
the product. Experimentally, the phase conjugate of
be obtained by phase conjugating one of the hologram writing
beam using a photorefractive crystal, such as BaTiO [40] or
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340
IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998
KNbO [41]. If
is an even function, then
is purely
. In this case, convolution will be
real and
the same as the correlation, as in our experiment, where the
square-wave packet is a symmetric function.
VI. CONCLUSION
We have demonstrated space-to-time conversion for generating femtosecond time-domain images which are the duplicates
of the space-domain images by using PRQW’s as dynamic
diffractive masks in a Fourier pulse shaper. We have shown
that the PRQW’s are suitable for Fourier-filtering applications,
and have also discussed image processing techniques for
both single and multiple images. A number of processing
operations, such as edge enhancement, Fourier transform,
and correlation, are realized by appropriately configuring the
hologram writing geometry and changing the power densities.
This will allow transfer of information at a high bandwidth and
bits of information to be packed together more tightly, which
could potentially result in higher data rate fiber lightwave
transmission systems.
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DING et al.: TIME-DOMAIN IMAGE PROCESSING USING DYNAMIC HOLOGRAPHY
Y. Ding received the B.S. and M.S. (with honors)
degrees from Tsinghua University, China, in 1985
and 1987, respectively, and the Dr. rer. nat. (Ph.D.)
degree from Technical University of Berlin, Berlin,
Germany, in 1992.
His thesis was on the photorefractive effects in
III–V and II–VI semiconductors, where he achieved
the largest two-wave mixing gain in InP:Fe at 1.06
m. From 1993 to 1996, he was a Research Scientist
in the Optical Institute at the Technical University
of Berlin, Berlin, Germany, and was a major contributor to several research programs supported by the Research Ministry of
the Federal Republic of Germany (BMFT), such as “optical bistability,” phase
conjugation,” “parallel switching of solid-state laser amplifiers,” with research
interests in photorefractive nonlinear optics, optical phase conjugation, semiconductor lasers and amplifiers. Since July 1996, he has been a Research
Associate at Purdue University, West Lafayette, IN. His current research
includes femtosecond time domain image processing, semiconductor MQW’s,
ultrafast and fiber optics. He has 80 journal publications and conference
presentations.
Dr. Ding is a member of the Optical Society of America and the New York
Academy of Sciences.
David Nolte was born in Akron, OH, on January 30,
1959. He received the bachelors degree in physics
from Cornell University, Ithaca, NY, in 1981, and
the Ph.D. degree in solid state physics in 1988
from the University of California at Berkeley for
research leading to the first experimental measurement of bandedge hydrostatic deformation potentials
in semiconductors using the concept of universal
reference levels.
In September of 1988, he joined Bell Laboratories
as a Post-Doctoral Member of technical staff. He
joined the faculty of Physics at Purdue University, West Lafayette, IN,
in 1989, where he was promoted to Associate Professor in 1994. His
research interests include ultralow-power nonlinear optics, optoelectronic
properties and applications of semi-insulating semiconductor heterostructures,
magnetic phenomena in electrooptics and nonlinear optics, and the physics of
mesoscopic point-like defects in semiconductors.
Dr. Nolte received the Presidential Young Investigator Award of the
National Science Foundation, and was awarded a Research Fellowship from
the Alfred P. Sloan Foundation. He has been elected a Fellow of the Optical
Society of America for the demonstration and development of the science and
applications of photorefractive semiconductor QW’s.
341
Michael R. Melloch (S’76–M’78–SM’91) received
the B.S.E.E., M.S.E.E., and Ph.D. degrees from
Purdue University, West Lafayette, IN, in 1975,
1976, and 1981 respectively.
From June 1976 to August 1978, he was a Design
Engineer at Intel Corporation where he worked on
the 8275, a CRT controller chip; the 8748, the first
single-chip microcomputer; and was codesigner of
the 8051, a second generation single-chip microcomputer. In February 1982, he joined the Central
Research Laboratories at Texas Instruments, Inc.,
as a member of the Technical Staff. At Texas Instruments, his research
interests centered around GaAs surface acoustic wave devices. In August
1984, he joined the School of Electrical Engineering at Purdue University as an
Assistant Professor and is presently a Full Professor there and former Assistant
Dean of Engineering. He has coauthored 275 conference talks, 210 technical
papers, 2 book chapters, and is holder of five U.S. patents. He was involved
with the first investigation of the effects of bandgap narrowing in GaAscaused by heavy impurity dopingon the electrical performance of devices and
the characterization of both the electron and hole minority carrier mobilities in
GaAs as a function of majority carrier concentration using a zero-field timeof-flight technique. He was involved with the development of a nonvolatile
memory cell in 6H-SiC; the development of the first digital integrated circuits
and CCD’s in SiC; the development of the first SiC DIMOS power transistor;
and the characterization and improvement of the SiO /SiC interface. He was
involved with the discovery of arsenic clusters in GaAs epilayers that were
grown at low substrate temperatures by MBE and subsequently annealed,
and the development of many electrical and optical applications of these
composite materials. In 1993, he co-founded MellWood Laboratories with
Prof. J. Woodall and Dr. E. Harmon. MellWood Laboratories is a supplier of
custom epitaxy and high-speed photodetectors.
Dr. Melloch has served as a Symposium Co-Organizer for the 1993 Spring
Meeting of the Material Research Society, the Program Chair for the 1994
North American Conference on Molecular Beam Epitaxy; Co-Organizer of
the Eight International Conference on Superlattices, Microstructures, and
Microdevices; and Associate Program Chair for the 1997 International Symposium on Compound Semiconductors. In 1993 he was elected to the Electronic
Materials Committee of TMS, and has served as Secretary (1995–1997) and
Vice-Chair (1997–1999). He has been guest editor for five journal special
issues. Prof. Melloch is a Fellow of the American Physical Society, and a
member the American Vacuum Society, the Minerals, Metals & Materials
Society, and the Optical Society of America.
Andrew M. Weiner (S’84–M’84–SM’91–F’95), for photograph and biography, see this issue, p. 331.
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